New generation energy pumps

1 Introduction

Among the rotary machines, high-pressure centrifugal pumps stand out, which are widely used in the power industry (Martsinkovsky, 2005; Gudkow et al., 2013). A characteristic stable trend is to increase the operating parameters: feeds, pressures and rotor rotation speeds, that is, to concentrate increasingly high powers in single units (Sayed et al., 2026; Marcinkovsky, 2012a). The pressure developed by centrifugal machines is proportional to the square of the rotor rotation frequency, so increasing the rotation frequency is the most rational way to achieve high pressures. As a result, high-pressure centrifugal machines, as a rule, are high-speed. And for such machines, the problems of rotor dynamics are especially relevant (Kundera and Marcinkowski, 2010; Pozovnyi et al., 2020).

The main source of vibrations is the rotor. The reasons for the unsatisfactory vibration state of the rotor can be imbalance and misalignment, loss of dynamic stability, structural and technological defects, wear of seals, supports, drive couplings (Pozovnyi et al., 2019; Tran and Haselbacher, 2004). Increased vibrations are accompanied by work near critical frequencies, the calculation of which, due to the lack of reliable data on the rigidity of the supports and the influence of many random factors, cannot guarantee reliable recovery from resonant modes (Marcinkovsky, 2012b; Youn et al., 2026).

Seals not only change the critical frequencies of the rotor, but also significantly affect the amplitudes of its forced oscillations and the limits of its stability (Nishimura et al., 2016; Ying et al., 2016). As a result, the problems of non-contact gap seals are intertwined with the problems of hydrostatic - hydrodynamic bearings and the problems of rotor dynamics (Yashchenko et al., 2017a; Yuan et al., 2024).

The rotor of a centrifugal pump rotating in gap seals represents a special closed hydromechanical system (Bai et al., 2013; Villa et al., 2008). The peculiarity of the system is due to the fact that hydrodynamic forces (and moments) of various nature act on the rotor from the side of the seals: inertial, dissipative, gyroscopic, potential and circulation. All these forces affect the rotor oscillations in different ways and, at the same time, depend on the nature of these oscillations (Ishida and Yamamoto, 2012; Verlan et al., 2023).

In the process of creating centrifugal pumps for any parameters, in addition to working out the economical flow part, the primary tasks are to reduce vibrations, ensure the necessary reliability and durability of supports and drive couplings, and develop reliable and sufficiently hermetic seals (Yashchenko et al., 2017b; Bird et al., 2025). The peculiarity of centrifugal machines is that the listed tasks are interrelated and, in most cases, can be satisfactorily solved by correctly choosing the designs of non-contact seals (Zhang et al., 2025; Liu and Hong, 2026).

The gap seals of the rotors have gaps of the same order as in the support bearings of the slide. Therefore, the seal is a full-fledged bearing, the bearing capacity of which is provided not only by the rotation of the eccentrically located shaft, but also, first of all, by a significant axial pressure drop, which is throttled on the seal (Liu et al., 2026; Gao et al., 2026). This hydrostatic component of the bearing capacity is of paramount importance, since, as a rule, it exceeds the bending stiffness of the shaft and the stiffness of radial plain bearings (Jin et al., 2026; Liu et al., 2024).

As a result, gap seals, in addition to their main purpose - to limit the flow between cavities with different pressures, can be used as rotor supports (Wang et al., 2013; Gorovoy and Golovchenko, 2023). In an implicit form, these functions of seals have always been implemented in centrifugal pumps.

The application of the computational environment for design parametric analysis transforms theoretical insights into a practical engineering tool (Balovsyak et al., 2019a). Through the use of virtual simulation techniques to process hydrodynamic data, it becomes possible to precisely identify the optimal range of the channel parameters that ensures rotor self-centering. Similar approaches to simulation-based analysis and structural parametric comparison have proven effective in complex engineering systems. This includes applications in the design of passenger rail car frames (Balovsyak et al., 2021), the study of strength in detachable transport modules, and the analysis of vehicle stability (Balovsyak et al., 2019b). Interactive visualization of amplitude-frequency characteristics enables engineers to predict system behavior across a range of operating conditions, effectively creating a digital representation of the seal assembly. This demonstrates that computational tools constitute an essential component of the data processing pipeline for investigating and managing complex physical phenomena in energy machinery, significantly reducing both the time and cost associated with the design phase.

Furthermore, advanced manufacturing processes for turbomachinery rotors are critical for minimizing static imbalance and the consequent reduction of vibration. In particular, the novel techniques introduced in (González-Barrio et al., 2020; Marin et al., 2001) demonstrate significant potential in this regard.

The research motivation is to harness the energy expended by the sealed fluid as it passes through the channels of non-contact seals—i.e., the pressure throttling within the clearances—for the purpose of suppressing vibration of the centrifugal-machine rotor. Based on this idea to substantiate the design of energy pumps in which the support functions are intentionally transferred to the gap seals.

2 Materials and methods

All sealing assemblies with throttling gaps or sealing paths filled with a high-pressure medium to be sealed should be considered dynamic systems. The sealing medium, acting on the walls of the sealing paths, affects the dynamic state of the rotor.

The basic principle of the approach described in (Shevchenko, 2025a) (Figure 1) is to use the energy lost in overcoming the throttling channels of the seals by the sealing medium to damp the kinetic energy of vibrations of the moving elements of centrifugal pumps, thereby improving their vibration characteristics. In this case, throttling through baffle seals, which throttle a huge pressure drop, can also play the role of dynamic supports.

The objective of the proposed method is to increase the service life and reliability of the centrifugal machine, expand the technological capabilities of non-contact rotor seals, which are simultaneously used as its dynamic supports. In particular, non-contact seals that throttle significant pressure drops can function as static supports, and with appropriate design - as dynamic supports. This consideration is important when designing critical power equipment.

The proposed methodology contributes to the creation of centrifugal pumps without auxiliary bearings, using non-contact seals on the impellers as rotor supports that operate on the pressure drop of the pumped liquid.

3 Results

3.1 Creation of pumps with bearings-seals

Examples of pumps in which contactless seals are used as rotor supports are shown in Figure 2. Figure 2a show a single-stage centrifugal pump of type C (console pump), the impeller of which is connected to a flexible drive shaft by means of a spherical spline connection. Figure 2b - a single-stage centrifugal pump of type D (with a double-entry wheel) - also with a flexible torsion. Figure 2c - a multistage sectional pump.

That is, the gap seals of the impellers, which operate due to the pressure difference of the pumped medium, play the role of not only static, but also dynamic supports, which, due to the pressure of the liquid being sealed, suppress rotor vibrations. Such pumps can operate without external supports. It is also visible that the seals of the sectional pump wheels are very densely placed along the rotor and due to their influence, the high-speed flexible rotor becomes rigid.

3.2 Parameters of seals-supports

Purposeful selection of seal design parameters allows to improve the vibration characteristics of the rotor (Hui et al., 2024). In such a configuration, a high-speed, dynamically flexible rotor in combination with appropriately designed seals effectively behaves as a “rigid” system. The proposed approach allows to use this effect depending on the selected seal design characteristics and, by changing them at the design stage, allows to adjust the “seal–rotor” system to avoid resonant operating conditions. Figure 3 shows a non-contact seal with a convergent (confusor) gap.

The performance of the gap between the rotor and the housing in the form of a confusor guarantees high radial rigidity and damping of the layer of the sealing medium in this gap, and hence the reliability and quality of the sealing unit. The confusing form of the sealing gap provides high rigidity of the layer of the medium, and hence its high centering ability. The energy of volume losses due to the seal in this case is used not only to ensure the required bearing capacity of the supports, but also to reduce the rotor vibration to an acceptable level even in the presence of significant imbalance. As can be seen from Figure 3b, the pressure distribution in the confusor gap is such that when the shaft is displaced relative to the housing, which leads to misalignment, a radial force arises that counteracts this displacement. This effect causes the rotor to self-center in the housing.

We introduce the concept of the channel taper parameterwhere ϑ 0 is total taper angle of the channel; l is channel length; H is average channel clearance.

θ 0 = ϑ 0 l / 2 H ,

3.3 Influence of seals on the centrifugal machines rotors dynamics

To illustrate the application of gap seals as load-bearing supports, Figure 4 presents a model of a single-stage centrifugal pump. In this configuration, the impeller (2) is coupled to a flexible drive shaft (9) via a spherical spline connection (10). Notably, the shaft diameter is determined solely by torsional strength requirements for transmitting the engine torque, without consideration for rotor dynamic stability. The radial position of the rotor is governed by the front seal (1) and the rear slotted seal element (5), which collectively function as hydrostatic radial bearings. Simultaneously, the axial position is regulated by a self-adjusting thrust hydrostatic bearing, realized through an automatic balancing system comprising an annular throttle (6), an end-face throttle (8), and an intervening pressure chamber (7).

The radial blades (9) in the housing (1) decelerate the circumferential flow in the rear sinus chamber (10), resulting in an increase in static pressure. This elevated pressure is subsequently throttled across the rear seal, thereby increasing the load-carrying capacity of the radial support (6). Owing to the flexibility of the shaft, the impeller retains freedom of radial, angular, and axial motion. Consequently, during operation, the impeller undergoes self-centering under the combined action of hydrodynamic forces and moments generated within the gap seals, together with the gyroscopic moment, ultimately assuming the most favorable neutral position with respect to external disturbances. As a result, the amplitudes of forced oscillations are reduced. At the drive-side end seal, the low inlet pressure is throttled, and due to the small diameter of the torsion bar, the seal operates at low peripheral velocities. These reduced operating demands contribute to an increase in service life.

The impeller of a shaftless pump equipped with a support–seal assembly constitutes an automatic control system for the variable end clearance, operating via negative feedback. Therefore, the dynamic behavior of this system can be investigated through the combined application of slotted seal models and automatic balancing device models (Shevchenko, 2025a). The impeller of a shaftless pump with a support–seal assembly functions as an automatic control system regulating the variable end clearance through negative feedback. Consequently, for the study of its dynamic characteristics, the models of slotted seals and automatic balancing devices can be applied jointly (Shevchenko, 2025a).

Since the non-contact operating mode of a shaftless pump is governed by the hydrodynamic characteristics of the automatic axial force balancing system and the slotted seals, the dynamics of its impeller, which floats freely within the slotted seals, must be examined.

The impeller, or rotor, of a shaftless pump possesses three degrees of freedom and undergoes coupled radial, angular, and axial vibrations. The coupling between radial and angular vibrations arises from hydrodynamic moments generated within the slotted seals. Radial and axial oscillations are interrelated through the dependence of annular throttle conductivities on eccentricity, i.e., on the radial displacement of the rotor axis relative to the housing axis.

As a preliminary step, angular vibrations of the impeller about its center of mass are considered independently of radial and axial displacements. Analysis of this simplified hypothetical model nevertheless provides a sufficiently comprehensive characterization of the principal features of angular oscillations within the broader dynamic system, which exhibits fully coupled radial, axial, and angular vibratory modes.

The equations of angular oscillations of a symmetrical single-disk rotor in two identical slotted seals were obtained in (Shevchenko, 2025b). In applying these equations to the present analysis, several simplifying assumptions are introduced:

• − the impeller without seals is assumed to have no angular stiffness;
• − angular oscillations are considered about the center of mass;
• − the resultant moment about the center of mass due to radial hydrodynamic forces arising in the slotted seals is taken to be zero;
• − the conductivity of the end throttle is assumed to be virtually independent of minor geometric distortions, so that the pressure in the chamber may be regarded as constant;
• − the impeller floats freely in the slotted seals, and in the equilibrium state the sum of all moments acting upon it is zero, meaning that the impeller performs free angular oscillations

Taking the foregoing simplifications into account, the initial equations take the following form:

b 1 θ ¨ x + b 2 θ ˙ x + b 3 θ x + b 4 θ ˙ y + b 5 θ y = 0 , b 1 θ ¨ y + b 2 θ ˙ y + b 3 θ y − b 4 θ ˙ x − b 5 θ x = 0 .

These equations take into account both the gyroscopic moment of the impeller and the hydrodynamic moments that occur in throat seals with a taper. These moments are characterized by coefficients b 1 − 5:

b 1 = 1 + k g 1 j 1 + k g 2 j 2 , b 2 = b 20 + b 21 , b 20 = c ϑ 0 I , b 21 = k d 1 j 1 + k d 2 j 2 ,

b 3 = − k p 1 θ 01 + N χ m 1 j 1 b * 1 + k p 2 θ 02 + N χ m 2 j 2 b * 2 ,

b * 1 , 2 = 10 χ m 1 , 2 θ 01 , 2 + χ m 1 , 2 ,

b 4 = b 40 + b 41 , b 40 = j 0 ω , b 41 = 0 , 5 k g 1 j 1 + k g 2 j 2 κ ω ,

b 5 = 0 , 5 k d 1 j 1 + k d 2 j 2 κ ω

The given coefficients have the following physical meaning: b 1 — coefficient of inertial moments; b 20 , b 21 — coefficients of external damping and damping in slotted seals; b 3 — coefficient of hydrostatic moments due to pressure drops throttled at the seals; b 40 — coefficient of the gyroscopic moment of the wheel; b 41 — coefficient of gyroscopic moments in the annular gaps; b 5 is the coefficient of circulation moments.

Additional coefficients for the self-similar region of turbulent flow are expressed by the formulas: m , I 0 , I — mass, polar and equatorial moments of inertia, R , l , H — radius, length, and radial clearance of the corresponding throat seal, θ x , y = ϑ x , y l / 2 H , ϑ x , y — angles of rotation about the axes Оx, Оy, c ϑ 0 — coefficient of external angular damping, к = 0.3 — twist coefficient flow in slotted seals, j 0 = I 0 / I , j = m l 2 / 60 I.

k g = ρ π R l 3 12 H m , k d = μ π R l 3 Λ 0 12 H 3 m , k p = Δ p π R l 2 H m , Λ 0 = 0 , 005 Re ,

Re = 2 ρ q μ , q = 10 Δ p H 3 ρ l 0 , 5 , N = 1 , χ m = ς 11 + ς 12 ς 11 − ς 12 + λ l / 2 H ,

ς 11 = 1 , 0 . . . 1 , 15 ; ς 12 = 0 , 05 . . . 0 , 3 ,

These equations take into account both the gyroscopic moment of the impeller and the hydrodynamic moments that occur in throat seals with a taper. These moments are characterized by coefficients b 1 − 5:

b 1 = 1 + k g 1 j 1 + k g 2 j 2 , b 2 = b 20 + b 21 , b 20 = c ϑ 0 I , b 21 = k d 1 j 1 + k d 2 j 2 ,

b 3 = − k p 1 θ 01 + N χ m 1 j 1 b * 1 + k p 2 θ 02 + N χ m 2 j 2 b * 2 ,

b * 1 , 2 = 10 χ m 1 , 2 θ 01 , 2 + χ m 1 , 2 ,

b 4 = b 40 + b 41 , b 40 = j 0 ω , b 41 = 0 , 5 k g 1 j 1 + k g 2 j 2 κ ω ,

b 5 = 0 , 5 k d 1 j 1 + k d 2 j 2 κ ω

The given coefficients have the following physical meaning: b 1 — coefficient of inertial moments; b 20 , b 21 — coefficients of external damping and damping in slotted seals; b 3 — coefficient of hydrostatic moments due to pressure drops throttled at the seals; b 40 — coefficient of the gyroscopic moment of the wheel; b 41 — coefficient of gyroscopic moments in the annular gaps; b 5 is the coefficient of circulation moments.

Additional coefficients for the self-similar region of turbulent flow are expressed by the formulas: m , I 0 , I — mass, polar and equatorial moments of inertia, R , l , H — radius, length, and radial clearance of the corresponding throat seal, θ x , y = ϑ x , y l / 2 H , ϑ x , y — angles of rotation about the axes Оx, Оy, c ϑ 0 — coefficient of external angular damping, к = 0.3 — twist coefficient flow in slotted seals, j 0 = I 0 / I , j = m l 2 / 60 I.

k g = ρ π R l 3 12 H m , k d = μ π R l 3 Λ 0 12 H 3 m , k p = Δ p π R l 2 H m , Λ 0 = 0 , 005 Re ,

Re = 2 ρ q μ , q = 10 Δ p H 3 ρ l 0 , 5 , N = 1 , χ m = ς 11 + ς 12 ς 11 − ς 12 + λ l / 2 H ,

ς 11 = 1 , 0 . . . 1 , 15 ; ς 12 = 0 , 05 . . . 0 , 3 ,

The amplitudes and phases of harmonic reactions, specifically the ratio of responses to individual harmonic excitations, are determined by the frequency transfer functions W j ω, which can be obtained by introducing the substitution p = j ω into the equation of axial vibrations:

W 1 i ω = u 1 A ¯ 1 ψ 1 = u 1 a A ¯ 1 ψ 1 a e i ϕ 1 = N 1 i ω D 0 i ω , W e i ω = u e a A ¯ e ψ e a e i ϕ e = N e i ω D 0 i ω , W ε i ω = u ε a A ¯ 2 ε a e i ϕ ε = k 3 D 0 i ω . (1)

We now separate the real and imaginary components of both the proper operator and the action operators:

D 0 = U 0 + i ω V 0 , U 0 ω = a 3 − ω 2 a 1 , V 0 ω = a 2 − ω 2 a 0 ;

N 1 = U 1 + i ω V 1 , U 1 = 1 − k 1 A 2 / A 1 , V 1 = T 2 ;

N e = U e + i ω V e , U e = 1 − k 2 A 2 / A e , V e = T 2 ; N ε = U ε = k 3 , V ε = 0 .

With these expressions substituted into Formula 1, the frequency transfer functions are expressed as complex numbers:where B 1 ω = u 1 a / A ¯ 1 ψ 1 a = W 1 i ω, ϕ 1 ω = arg ⁡ W 1 i ω is amplitude and phase frequency characteristics according to the impact ψ 1.

W 1 i ω = U 1 + i ω V 1 U 0 + i ω V 0 = u 1 a A ¯ ψ 1 a e i ϕ 1 = B 1 ω e i ϕ 1 ω ,

Separate the real and imaginary parts in the expression W 1:

Separating the real and imaginary parts in the expression for W 1 gives:

W 1 = U 1 + i ω V 1 U 0 + i ω V 0 = U 0 U 1 + ω 2 V 0 V 1 U 0 2 + ω 2 V 0 2 + i ω U 0 V 1 − U 1 V 0 U 0 2 + ω 2 V 0 2 .

The amplitude and phase of W 1 take the form:

B 1 ω = u 1 a A ¯ 1 ψ 1 a = U 1 2 + ω 2 V 1 2 U 0 2 + ω 2 V 0 2 , ϕ 1 ω = arctg ω U 0 V 1 − U 1 V 0 U 0 U 1 + ω 2 V 0 V 1 . (2)

We now determine the frequency characteristics with respect to the influences ψ e , ε:

B e ω = u e a A ¯ e ψ e a = U e 2 + ω 2 V e 2 U 0 2 + ω 2 V 0 2 , ϕ e ω = arctg ω U 0 V e − U e V 0 U 0 U e + ω 2 V 0 V e ; (3)

B ε ω = u ε a A ¯ 2 ε a = k 3 U 0 2 + ω 2 V 0 2 , ϕ ε ω = − arctg ω V 0 U 0 . (4)

According to the frequency characteristics, the critical rotor speeds can be determined: they correspond to the frequencies at which the amplitudes reach a maximum and the phases change by 180°.

In the first approximation, the rotor of a shaftless pump undergoes independent axial oscillations under kinematic excitation in the form of prescribed radial oscillations caused by the static imbalance of the impeller.

The dynamic stiffness of the controller then provides approximate estimates of the natural frequencies and the stability of axial oscillations. More accurate values of the critical frequencies are obtained from the amplitude and phase frequency responses.

For example, Figure 5 shows the frequency characteristics of the upgraded cantilever pump. It can be seen from the frequency characteristics that the first resonance occurs at a frequency of 610 s^–1. The second natural frequency of axial oscillations lies far beyond the limits of achievable rotational frequencies; therefore, it is of no practical interest.

Formulas 2–4 can be used to estimate the amplitudes of resonant oscillations. If the relative amplitudes of external disturbances are taken ψ 1 a = ψ e a = ε a = 0 , 1, then z 1 a = 0 , 098 z n = 0 , 024 m m ; z ε a = 0 , 034 z n = 0 , 009 m m .

The estimates obtained indicate that the critical frequency is twice the operating speed, meaning that the rotor is rigid with respect to axial vibrations. Even the resonant amplitudes corresponding to this unattainable critical speed do not pose a danger.

According to the Routh-Hurwitz criterion, for the stability of a third-order system, it is necessary and sufficient that the coefficients of the eigenoperator satisfy the inequality a 1 a 2 > a 0 a 3. Substituting the values of the coefficients, we reduce the stability condition to the form:

2 ς 2 + T 1 T 2 + κ s A ¯ 2 T 2 T 1 τ 2 T 2 2 ς + κ s A ¯ 2 τ 2 T 2 − 1 > 0 (5)

If the displacement time constant is greater than the compression time constant τ 2 > T 2, then Formula 5 is satisfied for any values of the damping index ς, including ς = 0. Using the values of constants, we obtain:

A 2 z 0 3 > V p n E Δ ψ 20 Δ ψ c 0 Δ ψ s 0 .

As a rule, the volume of the chamber V = A 2 H, where H is its depth. Therefore, in the first approximation, the condition of axial stability reduces to the inequality:

H < E z 0 3 p n · Δ ψ s 0 Δ ψ 20 Δ ψ c 0 , (6)

Where p n is nominal discharge pressure; and Δ ψ s 0 , Δ ψ 20 , Δ ψ c 0 are the dimensionless pressure drops in the chambers of the pump and off-loader.

Since the chamber depth H is an independent parameter that can be readily modified by design, condition (6) can be used when designing a pump to ensure its stability.

As can be seen from Formula 6, by adjusting the design parameters of the seals that are included in these inequalities, it is possible to expand the boundaries of stable operation.

When designing a shaftless pump, it is necessary to compute the axial and angular vibrations of the impeller freely floating in the slotted seals. This will make it possible to ensure the self-tightening of the rotor relative to its oscillations and to avoid destabilizing factors through a suitable selection of design parameters.

The angular oscillations of the free-floating impeller in slotted seals can be considered as independent oscillations. The hydrodynamic moments in the slotted seals and, above all, the gyroscopic moment of the impeller provide increased stiffness of the rotor with respect to its angular oscillations. Therefore, critical speeds of angular oscillations are usually absent. The main destabilizing factor is the circulating hydrodynamic moment, which is proportional to the average circumferential velocity of the fluid over the thickness of the gap.

Similar support-seals can also be used successfully in multistage centrifugal pumps.

4 Discussion

Numerical calculations were carried out for the rotor model with a disc between the seals. The gap seals with three taper parameters were considered: θ 0 = − 0 , 3 ; 0 ; 0 , 3. Diagrams for these parameters in Figure 6 are designated by numbers 1, 2, and 3, respectively.

The graphs clearly show that for the diffuser channel the total resistance is negative, that is, the amplitude of oscillations is maximum. For cylindrical and confusor channels the stability condition is fulfilled at all speeds. As can be seen from the example given, the geometry of the seal significantly affects the amplitude characteristics. For diffuser seals the critical frequency is much lower and the amplitude of oscillations is much higher than for confusor ones. The confusor shape of the seal suppresses the amplitude of oscillations and moves the critical frequency away.

As the study indicates, a taper parameter in the range of approximately 0.27–0.3 appears to be critical. This finding should be further validated through experimental investigations.

Indirect the critical taper ratio validation of the critical value was obtained through tests with the conical seal bushings installed in the reverse (diverging) configuration at a taper parameter of θ = -0.3. In this arrangement, shaft vibrations were observed even under static conditions, i.e., with no shaft rotation, when pressurized fluid was supplied to the sealed chamber. This observation provides qualitative support for the critical nature of the taper parameter identified in the numerical study.

Throttling in the confusor-type sealing gap, which is created between the rotor and the stator of a centrifugal machine, ensures the stability of the rotor motion at all studied speeds and an increase in the critical frequency of the rotor by 4–5 times compared to the rotor in smooth gaps. The use of the proposed method suppresses the amplitude of oscillations and moves the critical frequency away.

Thus, the analysis of dynamic characteristics showed that with a purposeful choice of geometric parameters of non-contact seals-supports, it is possible to change the oscillatory state of the centrifugal pump.

To date, experimental studies have been conducted on pumps without external radial bearings that operate with smooth annular seals. The results are presented in Figure 7 and confirm the feasibility of operating pumps without external radial bearings. Experimental investigations of support seals with convergent gaps are currently underway. Upon completion of the experiments, final recommendations will be provided and the agreement with the analytical findings will be assessed.

For successful implementation of the proposed method in pumping technology practice, laboratory tests of pumps with convergent-shaped bearing seals and industrial tests of modernized centrifugal pumps are envisaged.

5 Conclusion

The introduction of seals-supports with a convergent gap in the design of centrifugal pumps leads to a significant improvement in hydrostatic stiffness and damping, enhancing rotor dynamics. The results of the study predict a four-to five-fold increase in the critical speed, which is accompanied by a significant decrease in the amplitude of oscillations in the entire range of operating speeds. Non-contacting seals can function not simply as passive fluid retention elements, but as active stabilizers of the vibrational state of the pump. This is especially important for energy pumps with high specific power, operating under a wide range of discharge pressures and rotation modes.

The analytical basis laid in this work is a tool for the development of high-speed energy pumps of the next-generation, characterized by increased operational stability.

The analytical basis laid in this work is a powerful tool for the development of high-speed energy pumps of the next-generation, characterized by increased operational stability.

Promising areas of future research include:

• − improving the accuracy of developed hydromechanical models of rotor-seal systems by reducing discrepancies between numerical simulations and experimental data under nonlinear operating conditions and high rotational speeds;
• − developing and validating specialized experimental datasets for rotor-seal interaction systems, which allows for more reliable calibration and validation of analytical and numerical models that go beyond generalized assumptions;
• − investigating adaptive and optimized seal configurations, including variable-gap designs and modified geometry, to further improve vibration suppression and rotor stability under extreme pump operating conditions;
• – after the completion of experimental studies on innovative pump designs with various seal-support geometries will become possible substantiated recommendations for the application of modern manufacturing technologies.

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Footnotes

Edited by: Luis Norberto López De Lacalle, University of the Basque Country, Spain

Reviewed by: Abdul Wahab Hashmi, Shandong University of Technology, China

Jingyu Mo, Jimei University, China

Footnotes

A _*u*, amplitude of the rotor radial oscillations – m; e, eccentricity of mass center – m; F, hydrodynamic forces arising in sealing gap – N; Н, mean radial clearance of gap seal – m; h ₁, clearance at the entrance to the radial gap – m; h ₂, clearance at the exit from the gap – m; l, gap length – m; M, hydrodynamic moments arising in sealing gap – Nm; p ₁, pressure at the entrance to the gap – N/m²; p ₂, pressure at the outlet of the gap – N/m²; U _*i*, V _*i*, real and imaginary parts of differential operators; θ_*u* –amplitude of the rotor angular oscillations – rad; θ₀, taper parameter of the annular channel; ϑ ₀, radial taper of the annular channel – rad; ω, rotor speed – s⁻¹; ϕ u, ϕ θ, phases of radial and angular oscillations.